Solvable Set/Hyperset Contexts: III. A Tableau System for a Fragment of Hyperset Theory

We propose a decision procedure for a fragment of the hyperset theory, HMLSS, which takes inspiration from a tableau saturation strategy presented in [3] for the fragment MLSS of well-founded set theory. The procedure alternates deduction and model checking steps, driving the correct application of otherwise very liberal rules, thus significantly speeding up the process of discovering a satisfying assignment of a given HMLSS-formula or proving that no such assignment exists

Mapping Sets and Hypersets into Numbers

We introduce and prove the basic properties of encodings that generalize to non-well-founded hereditarily finite sets the bijection defined by Ackermann in 1937 between hereditarily finite sets and natural numbers

Sets as graphs

The aim of this thesis is a mutual transfer of computational and structural results and techniques between sets and graphs. We study combinatorial enumeration of sets, canonical encodings, random generation, digraph immersions. We also investigate the underlying structure of sets in algorithmic terms, or in connection with hereditary graphs classes. Finally, we employ a set-based proof-checker to verify two classical results on claw-free graph

CUMULATIVE HIERARCHIES AND COMPUTABILITY OVER UNIVERSES OF SETS

Various metamathematical investigations, beginning with Fraenkel’s historical proof of the independence of the axiom of choice, called for suitable definitions of hierarchical universes of sets. This led to the discovery of such important cumulative structures as the one singled out by von Neumann (generally taken as the universe of all sets) and Godel’s universe of the so-called constructibles. Variants of those are exploited occasionally in studies concerning the foundations of analysis (according to Abraham Robinson’s approach), or concerning non-well-founded sets. We hence offer a systematic presentation of these many structures, partly motivated by their relevance and pervasiveness in mathematics. As we report, numerous properties of hierarchy-related notions such as rank, have been verified with the assistance of the ÆtnaNova proof-checker.Through SETL and Maple implementations of procedures which effectively handle the Ackermann’s hereditarily finite sets, we illustrate a particularly significant case among those in which the entities which form a universe of sets can be algorithmically constructed and manipulated; hereby, the fruitful bearing on pure mathematics of cumulative set hierarchies ramifies into the realms of theoretical computer science and algorithmics.Various metamathematical investigations, beginning with Fraenkel’shistorical proof of the independence of the axiom of choice, called forsuitable definitions of hierarchical universes of sets. This led to the discovery of such important cumulative structures as the one singled out by von Neumann (generally taken as the universe of all sets) and Godel’s universe of the so-called constructibles. Variants of those are exploited occasionally in studies concerning the foundations of analysis (according to Abraham Robinson’s approach), or concerning non-well-founded sets. We hence offer a systematic presentation of these many structures, partly motivated by their relevance and pervasiveness in mathematics. As we report, numerous properties of hierarchy-related notions such as rank, have been verified with the assistance of the ÆtnaNova proof-checker.Through SETL and Maple implementations of procedures which effec-tively handle the Ackermann’s hereditarily finite sets, we illustrate a particularly significant case among those in which the entities which forma universe of sets can be algorithmically constructed and manipulated;hereby, the fruitful bearing on pure mathematics of cumulative set hierarchies ramifies into the realms of theoretical computer science and algorithmics

Set-syllogistics meet combinatorics

Peer reviewe

Set-syllogistics meet combinatorics

This paper considers 03* 00* prenex sentences of pure first-order predicate calculus with equality. This is the set of formulas which Ramsey's treated in a famous article of 1930. We demonstrate that the satisfiability problem and the problem of existence of arbitrarily large models for these formulas can be reduced to the satisfiability problem for 03* 00* prenex sentences of Set Theory (in the relators 08, =). We present two satisfiability-preserving (in a broad sense) translations \u3a6 \u21a6 (Formula presented.) and \u3a6 \u21a6 \u3a6\u3c3 of 03* 00* sentences from pure logic to well-founded Set Theory, so that if (Formula presented.) is satisfiable (in the domain of Set Theory) then so is \u3a6, and if \u3a6\u3c3 is satisfiable (again, in the domain of Set Theory) then \u3a6 can be satisfied in arbitrarily large finite structures of pure logic. It turns out that |(Formula presented.)| = (Formula presented.)(|\u3a6|) and |\u3a6\u3c3| = (Formula presented.)(|\u3a6|2). Our main result makes use of the fact that 03* 00* sentences, even though constituting a decidable fragment of Set Theory, offer ways to describe infinite sets. Such a possibility is exploited to glue together infinitely many models of increasing cardinalities of a given 03* 00* logical formula, within a single pair of infinite sets

Set Unification

The unification problem in algebras capable of describing sets has been tackled, directly or indirectly, by many researchers and it finds important applications in various research areas--e.g., deductive databases, theorem proving, static analysis, rapid software prototyping. The various solutions proposed are spread across a large literature. In this paper we provide a uniform presentation of unification of sets, formalizing it at the level of set theory. We address the problem of deciding existence of solutions at an abstract level. This provides also the ability to classify different types of set unification problems. Unification algorithms are uniformly proposed to solve the unification problem in each of such classes. The algorithms presented are partly drawn from the literature--and properly revisited and analyzed--and partly novel proposals. In particular, we present a new goal-driven algorithm for general ACI1 unification and a new simpler algorithm for general (Ab)(Cl) unification.Comment: 58 pages, 9 figures, 1 table. To appear in Theory and Practice of Logic Programming (TPLP

Decidability of E*A-sentences in Membership Theories

3The problem is addressed of establishing the satisfiability of prenex formulas involving a single universal quantifier, in diversified axiomatic set theories. A rather general decision method for solving this problem is illustrated through the treatment of membership theories of increasing strength, ending with a subtheory of Zermelo-Fraenkel which is already complete with respect to the There Exists*For All class of sentences. NP-hardness and NP-completeness results concerning the problems under study are achieved and a technique for restricting the universal quantifier is presented.opennoneopenOMODEO E.; PARLAMENTO F; POLICRITI A.Omodeo, E.; Parlamento, Franco; Policriti, Albert